1A distinction or difference between things. As a bare noun it sounds somewhat formal — most often appears in written-style phrases like 〜の別 (the distinction of X), 〜の別なく / 〜の別を問わず (regardless of X). In everyday speech, 違い or 区別 is more natural.
公私の別をしっかりつけることが大切だ。
It's important to draw a clear line between work and private life.
このイベントは老若男女の別なく、誰でも参加できます。
Anyone, regardless of age or gender, can take part in this event.
善悪の別がつかないうちは、まだ子供だと言える。
As long as someone can't tell right from wrong, you could call them still a child.
2Separate; different; another. Used in forms like 別の〜 (another / a different X), 別に (separately, or — colloquially — "not particularly"), 別々 (each separately). This is the most common everyday usage.
今日は忙しいので、別の日に会いましょう。
I'm busy today, so let's meet on another day.
それとこれとは別の問題だよ。
That and this are two different issues.
会計は別々でお願いします。
Could we get separate checks, please?
「怒ってる?」「別に。」
"Are you upset?" "Not really."
彼女には別に好きな人がいるらしい。
It seems she has someone else she likes.
3Exception; setting aside. Typically used in the patterns 〜は別として / 〜は別にして to set one item aside and talk about everything else ("setting X aside, …" / "X aside, …"). Common in writing and slightly formal speech.
値段は別として、品質はかなりいい。
Price aside, the quality is pretty good.
冗談は別として、これは本当に深刻な問題だ。
Joking aside, this is a really serious problem.
雨の日は別として、彼は毎朝ジョギングをしている。
Except for rainy days, he goes jogging every morning.
4Suffix attached to a noun to indicate classification by that category: 男女別 (by gender), 年齢別 (by age), 地域別 (by region), 国別 (by country), 価格別 (by price), etc. Extremely productive — used whenever data or items are grouped by some criterion.
売上を地域別にまとめてグラフにした。
I broke down sales by region and put the data in a graph.
このアンケート結果は年齢別に分析する必要がある。
These survey results need to be analyzed by age group.
商品が価格別に並んでいるので、予算で選びやすい。
The products are arranged by price, so it's easy to choose based on your budget.